Write the conjugate of the following complex number `(5i)/(7+i)`

Answer

To find the conjugate of the complex number \(\frac{5i}{7+i}\), we will follow these steps:

### Step 1: Identify the complex number
The complex number given is:
\[ z = \frac{5i}{7+i} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(7 - i\):
\[ z = \frac{5i}{7+i} \cdot \frac{7-i}{7-i} = \frac{5i(7-i)}{(7+i)(7-i)} \]

### Step 3: Simplify the denominator
Using the difference of squares formula, we simplify the denominator:
\[ (7+i)(7-i) = 7^2 - i^2 = 49 - (-1) = 49 + 1 = 50 \]

### Step 4: Simplify the numerator
Now, we simplify the numerator:
\[ 5i(7-i) = 35i - 5i^2 \]
Since \(i^2 = -1\), we have:
\[ -5i^2 = 5 \quad \text{(because } -5 \cdot (-1) = 5\text{)} \]
Thus, the numerator becomes:
\[ 35i + 5 = 5 + 35i \]

### Step 5: Combine the results
Now, we can write \(z\) as:
\[ z = \frac{5 + 35i}{50} \]

### Step 6: Write the complex number in standard form
This can be expressed as:
\[ z = \frac{5}{50} + \frac{35i}{50} = \frac{1}{10} + \frac{7i}{10} \]

### Step 7: Find the conjugate
The conjugate of a complex number \(z = x + yi\) is given by \(z^* = x - yi\). Therefore, the conjugate of \(z\) is:
\[ z^* = \frac{1}{10} - \frac{7i}{10} \]

### Final Answer
Thus, the conjugate of the complex number \(\frac{5i}{7+i}\) is:
\[ \frac{1}{10} - \frac{7i}{10} \]

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